# ABOUT THE PROGRAMS The undergraduate REU and SPUR Programs will be in-person

The REU and SPUR summer programs provide the opportunity for undergraduate students of mathematics to participate in leading-edge research.  Projects are designated "REU" and others are designated "SPUR".  The difference between REU and SPUR projects is the funding support that is available (See Support/Costs below).

REU PROGRAM - Research Experience for Undergraduates
SPUR PROGRAM - Summer Program for Undergraduate Research

The Nexus Scholars Program
The Math Department REU and SPUR programs will also be inviting students from the Cornell College of Arts And sciences, Nexus Scholars Program to work alongside participants in the REU and SPUR Programs.  This is a new program offered and administered by the Cornell University College of Arts and Sciences that provides Cornell undergraduate students the opportunity to work on research projects side-by-side with faculty members across the college including the Math Department.  Cornell students interested in the NSP can go to the Nexus Scholars Program website for more information on the program and information on how to apply.

PROGRAM DATES AND LOCATION:
Both the REU and SPUR Programs are 8-week programs that will run from
June 6, 2022 - July 29, 2022
The Nexus Scholars Program runs from May 31, 2022 - July 22, 2022
The location of the REU and SPUR programs will be held in the Department of Mathematics, Malott Hall, Cornell University

# REU and SPUR PROGRAM PROJECT DETAILS:

REU Project 1 Title:  Periodic non-intersecting random walks and random matrices
Directed by:  Andrew Ahn
Project Details:  The project is focused on a non-intersecting random walk model on a cylinder with connections to a variety of statistical mechanics models.  The marginals of this random walk can be interpreted as a random matrix model with a bias on the eigenvectors, known as the Moshe-Neuberger-Shapiro (MNS) model.  It can also be realized as a diffuse limit of lozenge tiling models of the cylinder.  The goal of the project is to find limit shapes for his random walk and describe the fluctuations from the limit shape.  The fluctuations are believed to relate to th Gaussian free field, a 2 dimensional random field which appears in a variety of statistical mechanics model, with some additional features inherited from the geometry of the model.  The MNS model possesses a rich algebraic structure due to its connection with symmetric functions, which will serve as an entry point for analysis.

REU Project 2 Title:  Nonintegrable Constraints in Mechanics
Directed by:  William Clark
Project details:  Newton's laws state that a body subjected to zero force will experience zero acceleration.  A geometric interpretation of zero acceleration is straight line motion.  This observation leads to Lagrangian mechanics where physical motion satisfies the principle of least action - straight lines minimize the distance between two points and physical motion, analogously, minimizes the action.

Complexities can be introduced by imposing constraints:  the bob on a pendulum must remain a fixed distance from the pivot, a bicycle cannot move sideways, and a billiard ball must remain within the confines of the tabletop.  The first constraint only depends on the position of the pendulum (holonomic), the second depends on the velocity of the bike (nonholonomic), and the third is an inequality constraint on the position of the ball (unilateral).  One particularly strange facet of nonholonomic systems is that they fail to obey the principle of least action.  Students participating in this project will investigate properties of nonholonomic and unilaterally constrained systems and how they inter-relate.

Students for this program should have a strong understanding in linear algebra, exposure to qualitative reasoning of ordinary differential equations, and good computer programming skills.  Background in the following areas will be helpful but not required:  Newtonian/Lagrangian/Hamiltonian mechanics and exposure to manifolds.

REU Project 3 Title:  Optimality & Uncertainty
Project details:  Equations describing optimal behavior often present serious computational challenges.  (The quickest driving directions?  The most energy-efficient trajectory for a Mars-rover?  The risk-of-detection-minimizing flight-plan for a spy plane?  The best schedule for a drug therapy for cancer patients?  The best strategy for a predator to hunt its prey?)  The need for efficient algorithms becomes particularly obvious once you add to the mix the uncertainty about our environment, conflicting goals, and multiple (competing or cooperating) participants.  Students participating in this project will investigate the theoretical properties and build fast algorithms for optimal control and differential games.  We will also explore the usefulness of such algorithms for problems arising in robotic navigation, traffic engineering, environmental crime modeling, ecological management, and design of adaptive drug therapies.  Successful candidates will need good programming skills, previous exposure to ordinary differential equations and numerical computing.  Some background in the following areas will be also helpful, but is not expected or required:  partial differential equations, game theory, probability theory.  More on this project can be found at

SPUR Project 4 Title:  Symplectic embeddings of 4D toric domains
Directed by:  Morgan Weiler
Project details:  Our project will focus on symplectic embeddings of toric domains, which are regions in four-dimensional space equipped with a "symplectic form", a gadget which generalizes the concept of conservation energy.  Symplectic embeddings of 4D domains encode coordinate transformations of the phase spaces of 2D physical systems.  These problems are very hands-on, and students will be able to use cutting-edge geometric tools via combinatorial and number-theoretical methods.  We are committed to making this an inclusive experience, and students from all backgrounds are encouraged to apply.

Prerequisites:  At least one proof-based mathematics course.  Applications from students who have taken (or will take in Spring 2022) at least one proof-based geometry or topology course will be given preference.  If students have prior experience with mathematical software (e.g. Mathematica), that will be useful, but is not necessary.  The most important thing is an interest in solving geometrically motivated problems via hands-on methods.  If you have questions regarding the project, please feel free to contact Morgan Weiler.

# OTHER IMPORTANT DETAILS

SUPPORT/COSTS:

• The REU projects are supported by the NSF (https://www.nsf.gov/) RTG Grant.  Students who are U.S. citizens or permanent residents are eligible to apply.  If selected students will receive funding support of $5,000.00 under this NSF grant. • Any student funding that is provided for the SPUR program comes from the Cornell Department of Mathematics. To receive Cornell funding for SPUR you must be a Cornell student, but do not need to be a US citizen or resident. Non-Cornell students are welcome to apply and participate in the program, but there is no funding available. • Cornell students selected for either the REU or SPUR Program may apply for departmental support of$3,500.00.
• Funding information for the Nexus Scholars Program is found at https://as.cornell.edu/research/nexus-scholars

We welcome applications from all students including Cornell, non-Cornell, and international students (see below under International Students).  Those who apply to the programs as self-funded applicants will still be subject to the same competitive selection process.  If you are not eligible for REU funding, you can still apply to any of the three REU program projects as a self-funded applicant.

HOUSING:

Participants will arrange for their own housing.  The Math Department will provide information on housing and contact information for participants to reserve their housing.  Depending on the impacts of Covid housing may be restricted and limited to external students in the affordable on campus Coop housing https://scl.cornell.edu/residential-life/housing/campus-housing/upperlevel-undergraduates/cooperative-housing , which is the preferred choice for program participants and is generally where program participants have resided in previous years.  We are currently working with Cornell Conference Services on options for on-campus dorm housing.

AFTER YOU ARE SELECTED:

After you are selected, you will be registered as a Cornell Student as follows:

1. If you are accepted to either program and you are a student (US or non-US/International student) currently attending and enrolled at a U.S. academic institution in a full-time 4-year undergraduate degree program, you will be registered through Cornell Summer Session, and no fees apply.  At the end of the program, you will receive an “S” for satisfactory completion of the program.

International students with their home institution located at a non-U.S. college/university are also registered through Cornell Summer Session but under the umbrella of the IRIP Program (International Research Internship Program - https://international.globallearning.cornell.edu/host-departments/international-research-intern-program).  We welcome international students to apply!  The IRIP Program only applies to full-time international students who are currently registered and attending non-U.S. academic institutions full-time and are coming directly from their international institution.  The IRIP Program also applies if you are an international student currently registered and attending a non-U.S. institution full-time enrolled in an exchange program and conducting academic work at a U.S. institution.  International students accepted to the REU or SPUR programs will be self-funded, enroll through Summer Session under the umbrella of IRIP, and will be required to pay an administrative fee.  In addition, you will be required to have or obtain adequate health insurance that is university compliant.  Under the IRIP Program you will earn 6 credits - Satisfactory or Unsatisfactory (but not a letter grade).

IRIP Costs:  The student is responsible for paying an administrative fee of approximately $2,430.00 for summer 2022. IRIP offers a scholarship that is 25% of the current summer extramural tuition rate for 6 credits. The administrative fee is determined as follows: Tuition cost per credit x 25% x 6 credits = administrative fee. These will not be graded credits. HEALTH INSURANCE REQUIREMENTS FOR PARTICIPANTS: There is a small health fee that is charged for a basic plan through Cornell Health, which is covered by the Math Department. The plan covers any basic care that may be needed during your time at Cornell. Universities and colleges in the U.S. require all students to have adequate and comprehensive health insurance, and any current plan a student has at their home U.S. based college should be compliant. This also includes any U.S. based insurance plan that a student may have under their parent/guardian policy. However, you should check that there is no lapse in coverage and that your plan will cover healthcare while you are participating in the program. Cornell health insurance requirements can be found at https://studenthealthbenefits.cornell.edu/requirements/summer-students, and you have the option to purchase Cornell SHP if you choose. The program SHP cost for June 2022 is$285.00.  The rate for July is still being developed, and information will be available at the end of February.  The estimated cost will be in the range of $600.00. If you are a participant that falls under the IRIP Program you will be required to have or obtain health insurance that is comprehensive and compliant with Cornell guidelines and is provided by a company licensed to do business in the U.S. You also have the option to purchase the above-mentioned Cornell SHP. The same rate applies. COSTS/EXPENSES: • IRIP Program Administrative Fee = 2,430.00 - This only applies to participating international students who fall under the IRIP Program. • Health Insurance (Cornell Student Health Plan, SHP) =$600.00, estimated for 2022.  Applies to students who fall under the IRIP Program.  Note: all participants will automatically receive basic care coverage with health fee applied that is covered by the department (explained above).  Students who want a more comprehensive coverage that do not have a current comprehensive health plan can purchase the SHP.
• Housing/living costs (applies to all REU/SPUR students) – If you are able to secure Coop housing, rates range from approximately $70.00 -$130.00/week depending single/double occupancy and where you stay.  As mentioned previously, Coop housing will likely be limited for Summer 2022, and current Coop members will be given priority.  Other on-campus dorm housing is available, but at a higher cost.  The cost is $39.00 per person per night for a double room and$48.00 per person per night for a single room.  More information on housing and Coop availability will be available in early 2022.  There is also the option of other off-campus housing including sublets, which becomes available during the summer months.

• Participants can also purchase meal plans and Parking Plans through Cornell Dining and Cornell Transportation Services.  Some housing at the Coops has free parking.

• Budget for food and other miscellaneous living expenses.

# TO APPLY

To apply for the REU and SPUR Programs you will need the following:

1. Statement about your background, educational goals, and your scientific interests.  Include whatever further information you consider relevant and be sure to include information about your computer experience.
2. At least two letters of recommendation, which can be uploaded to the application portal.  The reference writer is notified to upload their letter once you enter their information on the application.
3. Transcripts (unofficial transcripts are accepted), which can be uploaded to the application portal.
The application link will be available the first week in January
If you have comments, questions, or concerns please send e-mail to the SPUR and REU coordinators at mathreu@cornell.edu.

Please apply by going to Mathprograms.org, and go to Program ID #1216.  The deadline to submit your application for the REU and/or SPUR Programs is February 26, 2022.  Offers will be made approximately March 10, 2022.

Cornell students who want to apply for the Nexus Scholars Program in the Cornell College of Arts and Sciences to the Nexus Scholars Program website.

If you have comments, questions, or concerns please send e-mail to the SPUR and REU coordinators at mathreu@cornell.edu

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